Playing with infinity: the basketball, part 2 (part 1 solved)

[bonanova]

An idealized basketball falling from a height h bounces from the floor to a height h/2.

Tell us two things:

1. The ball is dropped from a height of 1m.
   Does it come to rest (stop bouncing) in finite time?
   
2. Xavier, in shows us that after 9.31 seconds, the ball comes to rest.
   
   We now specify that the ball is blue initially and on each bounce it changes color,
   alternating between blue and red. After the ball comes to rest, it ceases to change color.
   
   Question 2: what is its color after coming to rest?

Edited June 13, 2013 by bonanova
Pose Part 2.

[Yoruichi-san]

Blue, I believe. It takes 115.57 bounces to reach the Planck length so after the 116th bounce it should stop moving.

[Barcallica]

If bounce is ideal, it will never stop, be it started from 1m height or or 0.5m no difference

[James33]

Assuming that just the basics of mechanics apply (no compression of the ball or the ground, no stickyness of the ground etc. then it will never stop.

[Quantum.Mechanic]

Please define "ideal". To me, "ideal" means the ball would bounce back to height h (assuming various things like the floor is the frame of non-rotating reference, perfectly elastic collision, etc.)

Since you state the ball bounces back to h/2, something else must be going on. For instance, the ball at h could be going faster than terminal velocity in air a . Or the floor is not a perfectly elastic surface.

[Xavier]

Assuming the Ball instantly loses half its velocity when it hit the ground:

Zeno

A ball on the ground boucing at Velocity V will go up and down in t=2V/g seconds

next bound will start with V/2, the next, V/4 etc..

So the total bounces time for the infinite number of bounces is T=2V/g*sum(n in 0..inf , 1/2^n)

the sum converge to 2

to the time of bounce is finite and is 4V/G (plus the initial time for the ball to all from h).

As the problem stated, H=1m so it take Sqrt(2/g) to hit the ground and will have a speed of sqrt(2g). V is half of that

So T=sqrt(2/g)+4*sqrt(2g)/2 = 9.31s (with g=9.81)

Edited June 12, 2013 by Xavier

[bonanova]

Please define "ideal". To me, "ideal" means the ball would bounce back to height h (assuming various things like the floor is the frame of non-rotating reference, perfectly elastic collision, etc.)

Since you state the ball bounces back to h/2, something else must be going on. For instance, the ball at h could be going faster than terminal velocity in air a . Or the floor is not a perfectly elastic surface.

By elastic I would mean returning to the full height h. This ball is not elastic.

By ideal I mean to say neglect air resistance, energy loss to acoustic processes, energy lost to deformation of the ball that might depend on impact velocity, and so on. That is, this idealized ball has none of these bothersome second-order details; it has a behavior that is completely described for the purpose of the puzzle as "returning to half the height, on each bounce, from which it fell."

Aside from this, the laws of physics, in particular a constant acceleration due to gravity, apply.

[bonanova]

Assuming the Ball instantly loses half its velocity when it hit the ground:

Zeno

A ball on the ground boucing at Velocity V will go up and down in t=2V/g seconds

next bound will start with V/2, the next, V/4 etc..

So the total bounces time for the infinite number of bounces is T=2V/g*sum(n in 0..inf , 1/2^n)

the sum converge to 2

to the time of bounce is finite and is 4V/G (plus the initial time for the ball to all from h).

As the problem stated, H=1m so it take Sqrt(2/g) to hit the ground and will have a speed of sqrt(2g). V is half of that

So T=sqrt(2/g)+4*sqrt(2g)/2 = 9.31s (with g=9.81)

Zeno indeed is watching this ball.

Not every infinite series converges, but the infinite sequence of bounce times in this case does sum to a finite number.

According to Xavier, 9.31 seconds after the ball is dropped, it comes to rest.

Part 1 of the basketball puzzle is solved.

The OP has been edited to include Part 2.

[James33]

An idealized basketball falling from a height h bounces from the floor to a height h/2.

Tell us two things:

1. The ball is dropped from a height of 1m.

   Does it come to rest (stop bouncing) in finite time?

2. Xavier, in shows us that after 9.31 seconds, the ball comes to rest.

   We now specify that the ball is

   blue initially and on each bounce it changes color,

   alternating between

   blue and red. After the ball comes to rest, it ceases to change color.

   Question 2: what is its color after coming to rest?

As the bouces quicken, it changes between blue and red quicker and quicker so when it stops bouncing it is changing between them infinitly quickly so its clolour will be purple.

[bonanova]

An idealized basketball falling from a height h bounces from the floor to a height h/2.Tell us two things:

• The ball is dropped from a height of 1m.

  Does it come to rest (stop bouncing) in finite time?

• Xavier, in shows us that after 9.31 seconds, the ball comes to rest.

  We now specify that the ball is

  blue initially and on each bounce it changes color,

  alternating between

  blue and red. After the ball comes to rest, it ceases to change color.

  Question 2: what is its color after coming to rest?

As the bouces quicken, it changes between blue and red quicker and quicker so when it stops bouncing it is changing between them infinitly quickly so its clolour will be purple.

And after it stops bouncing?

[TimeSpaceLightForce]

Is Planck's length to be considered?

[bonanova]

Is Planck's length to be considered?

Not for idealized basketballs.

However, if we stop the bouncing at Plank's height, you could calculate the final color.